Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models.The specific diagnostic tool used to study the crossovers is the level spacing distribution.The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin-orbit coupling.We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place.This exponent is independent of microscopic details of the perturbation.We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution.We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.
Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models.The specific diagnostic tool used to study the crossovers is the level spacing distribution.The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin-orbit coupling.We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place.This exponent is independent of microscopic details of the perturbation.We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution.We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.
Using numerical diagonalization we study the crossover among different random matrix ensembles (Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE) and Gaussian symplectic ensemble (GSE)) realized in two different microscopic models. The specific diagnostic tool used to study the crossovers is the level spacing distribution. The first model is a one-dimensional lattice model of interacting hard-core bosons (or equivalently spin 1/2 objects) and the other a higher dimensional model of non-interacting particles with disorder and spin-orbit coupling. We find that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We also find that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. We also conjecture that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system.
In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions
, (2) studied the phenomenon of many-body localization in a model with a single
particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the
application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random
matrix ensembles. We conclude with a brief summary of each chapter.
In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking
perturbation. The crossover value of the perturbation scales as a power law
∼ L−3 when the integrable system is gapless and the scaling appears to be robust to
microscopic details and the precise form of the perturbation.
In Chapter 3, we have studied the crossover among different random matrix ensembles
CHAPTER 6. CONCLUSION 127
[Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE)
and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We
have found that the perturbation causing the crossover among the different ensembles
scales to zero with system size as a power law with an exponent that depends on the
ensembles between which the crossover takes place. This exponent is independent of
microscopic details of the perturbation. We have also found that the crossover from the
Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover
which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE
associated with symmetry breaking introduces a subdominant contribution. Finally,we
have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the
dimensionality of the system.
In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson
localized systems. These conservation laws are found as power series in the hopping
parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using…
We study the spectral properties of and spectral crossovers between different random matrix ensembles [Poissonian, Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE)] in correlated spin-chain systems, in the presence of random magnetic fields, and the scalar spin-chirality term, competing with the usual isotropic and time-reversal invariant Heisenberg term. We have investigated these crossovers in the context of the level-spacing distribution and the level-spacing ratio distribution. We use random matrix theory (RMT) analytical results to fit the observed Poissonian-to-GOE and GOE-to-GUE crossovers, and examine the relationship between the RMT crossover parameter $\ensuremath{\lambda}$ and scaled physical parameters of the spin-chain systems in terms of a scaling exponent. We find that the crossover behavior exhibits universality, in the sense that, it becomes independent of lattice size in the large Hamiltonian matrix dimension limit. Moreover, the scaling exponent obtained from such a finite size scaling analysis, seems to be quite robust and independent of the type of crossover considered or the specific spectral correlation measure used.
We study models of interacting fermions in one dimension to investigate the crossover from integrability to nonintegrability, i.e., quantum chaos, as a function of system size. Using exact diagonalization of finite-sized systems, we study this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size $L\ensuremath{\rightarrow}\ensuremath{\infty}$ nonintegrability sets in for an arbitrarily small integrability-breaking perturbation. The crossover value of the perturbation scales as a power law $\ensuremath{\sim}{L}^{\ensuremath{-}\ensuremath{\eta}}$ when the integrable system is gapless. The exponent $\ensuremath{\eta}\ensuremath{\approx}3$ appears to be robust to microscopic details and the precise form of the perturbation. We conjecture that the exponent in the power law is characteristic of the random matrix ensemble describing the nonintegrable system. For systems with a gap, the crossover scaling appears to be faster than a power law.
We investigate the spectral properties of all-to-all interacting spin Hamiltonians acting on exactly k spins, whose coupling coefficients are drawn from a normal distribution with mean μ and variance σ2. For μ=0, we demonstrate that the random matrix ensemble -- Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE), or Gaussian Symplectic Ensemble (GSE) -- is determined by the parity of system size L and locality k, following standard time-reversal symmetry classification. For couplings with a nonzero mean, we map the Hamiltonians to deformed random matrix ensembles and analyze conditions for an energy gap between the ground state and the first excited state. For μ<0, we find two distinct regimes: for k≫L, the gap closes at critical disorder σc≈∣μ∣. Near this transition the energy gap Δ exhibits universal quadratic scaling Δ/L∼(σ−σc)2. When k≪L, σc scales with ∣μ∣, but lacks a sharp transition. Our work introduces a semi-solvable model that captures universal features of random-matrix statistics, and spectral gap formation, providing a foundation for systematic extensions to more general many-body systems.
We investigate several distinct spectral crossovers amongst various integrable (Poissonian) and quantum-chaotic (Wigner-Dyson) limits of a 1D disordered quantum spin (S=1/2) model, by tuning the relative amplitudes of various Hamiltonian parameters to retain or break relevant unitary and antiunitary symmetries. The spin model consists of an isotropic and deterministic Heisenberg term, a random Ising term, an anisotropic and antisymmetric, but deterministic Dzyaloshinskii-Moriya (DM) term, and a Zeeman coupling to a random, inhomogeneous magnetic field. Since we are specially interested in crossovers involving a Gaussian symplectic ensemble (GSE) limit, we carry out all our calculations with an odd number of lattice sites (spins) that naturally results in eigenspectra with Kramers degeneracies (KD's). The various crossovers (viz., the reentrant Poissonian-to-GSE-to-Poissonian, Poissonian-to-GUE, GSE-to-GUE and the reentrant Poissonian-to-GOE-to-Poissonian crossovers) are investigated via detailed studies of both short-range (nearest-neighbour-spacings distribution, NNSD) and long-range [spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}(L)$ and number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}(L)$ ] spectral correlations, where L is the spectral interval over which the long-range statistic is examined. The short-range studies show excellent agreement with RMT predictions. One of the highlights of this study is the systematic investigation of the consequences of retaining both eigenvalues corresponding to every Kramers doublet, in a crossover involving the GSE limit, and see how it evolves to a limit where the KD is naturally lifted. This is seen most clearly in the NNSD study of the GSE-to-GUE (Gaussian unitary ensemble) transition, achieved by gradually lifting the KD, using the random magnetic field. The NNSD plot in the GSE limit here exhibits a Dirac delta peak at zero splitting and a renormalized GSE hump at finite splitting, whose general analytical form and its asymptotic limit are derived. With an increasing symmetry breaking magnetic field the NNSD shows an interesting, dynamic two-peaked structure that finally converges to the standard GUE lineshape. We explain this trend in terms of a competition between the splittings amongst distinct Kramers doublets (related to unitary symmetries) and the Zeeman-like splittings induced by a breaking of the antiunitary time-reversal symmetry (TRS). This is investigated via the NNSD, the marginal spectral density (MSD) and the densities of states (DOS) for both spin models and RMT crossover matrix models. The first and the final short-range studies involve reentrant Poissonian-to-GOE(GSE)-to-Poissonian crossovers, where the final Poissonian is obtained by a many-body localization of states in the strongly disordered limit, whereas the initial Poissonian regime involve much more delocalized eigenstates. In the long-range spectral correlation studies, we shed light on the extent of agreement between our physical spin systems and RMT predictions. We find that the spin systems depart from the ideal RMT predictions for relatively finite $L\ensuremath{\sim}10\ensuremath{-}15$ at least, for the spectral rigidity and a much smaller $L\ensuremath{\sim}2\ensuremath{-}4$ for the number variance. It is further seen that the departure is usually sooner at the uncorrelated (Poissonian) upper end compared to the correlated (Wigner-Dyson) lower end. We carry out a detailed comparison between the local and the global crossover points, associated with the short-range and the long-range statistics respectively, and find that in most cases they seem to agree reasonably well, but for a few exceptions. Our studies also show that the long-range correlations may serve to distinguish between the two Poissonian limits (nonlocalized and localized) in the reentrant crossovers, which the short-range correlations fail to distinguish.